Question

Divide the following and write your answer in lowest terms: $\dfrac{{{x}^{2}}-4x-5}{{{x}^{2}}-25}\div \dfrac{{{x}^{2}}-3x-10}{{{x}^{2}}+7x+10}$.
A. $\dfrac{x+1}{x-5}$
B. $\dfrac{x-1}{x-5}$
C. $\dfrac{x+1}{x+5}$
D. None of these

Answer 1

Hint: First we use middle term splitting for the factorization of each and every expression in the question. After that divide each and every expression as per given in the question. By this we can easily get the answer which would be correct.

Complete step-by-step answer:

$\dfrac{{{x}^{2}}-4x-5}{{{x}^{2}}-25}\div \dfrac{{{x}^{2}}-3x-10}{{{x}^{2}}+7x+10}$
Now, we have to solve the equation ${{x}^{2}}-4x-5$, by using middle term splitting. By this we get the factors of the equation.
$\begin{align}
  & \Rightarrow {{x}^{2}}-4x-5 \\
 & \Rightarrow {{x}^{2}}+x-5x-5 \\
 & \Rightarrow x\left( x+1 \right)-5\left( x+1 \right) \\
 & \Rightarrow (x+1)(x-5)\ldots (1) \\
\end{align}$
$\therefore $The factors of ${{x}^{2}}-4x-5$ is $(x+1)(x-5)$ …………………….(1)
Now, we have to solve ${{x}^{2}}-25$ by using the identity $\therefore {{a}^{2}}-{{b}^{2}}=\left( a+b \right)\left( a-b \right)$. By this we get the factors of the equation.
${{x}^{2}}-25=(x-5)\left( x+5 \right)....(2)$
$\therefore $The factor of ${{x}^{2}}-25$is $(x+5)(x-5)$ …………………………..(2)
After that, we divide equation (1) and equation (2)
$\Rightarrow \dfrac{\left( x+1 \right)\left( x-5 \right)}{\left( x+5 \right)\left( x-5 \right)}$
The reduced form of the above expression is:
$\Rightarrow \dfrac{(x+1)}{(x+5)}....(3)$

Now, considering the expression ${{x}^{2}}-3x-10$, we simplify the equation using middle term splitting. By this we get the factors of the equation,
$\begin{align}
  & \Rightarrow {{x}^{2}}-3x-10 \\
 & \Rightarrow {{x}^{2}}+2x-5x-10 \\
 & \Rightarrow x\left( x+2 \right)-5\left( x+2 \right) \\
 & \Rightarrow \left( x+2 \right)\left( x-5 \right) \\
\end{align}$
$\therefore $The factors of ${{x}^{2}}-3x-10$is $(x+2)(x-5)$ ………………..(4)
Now, Equation ${{x}^{2}}+7x+10$, we simplify the equation using middle term splitting. By this we get the factors of the equation,
$\begin{align}
  & \Rightarrow {{x}^{2}}+7x+10 \\
 & \Rightarrow {{x}^{2}}+2x+5x+10 \\
 & \Rightarrow x(x+2)+5(x+2) \\
 & \Rightarrow (x+2)(x+5) \\
\end{align}$
$\therefore $The factors of \[{{x}^{2}}+7x+10\] is $(x+2)(x+5)$ ……………………(5)
Further, Divide equation (4) and equation (5)
$\begin{align}
  & \Rightarrow \dfrac{\left( x+2 \right)\left( x-5 \right)}{\left( x+2 \right)\left( x+5 \right)} \\
 & \Rightarrow \dfrac{\left( x-5 \right)}{(x+5)}....(6) \\
\end{align}$
Divide equation (3) and equation (6)
\[\begin{align}
  & \Rightarrow \dfrac{\left( x+1 \right)\left( x+5 \right)}{\left( x+5 \right)\left( x-5 \right)} \\
 & \Rightarrow \dfrac{\left( x+1 \right)}{\left( x-5 \right)} \\
\end{align}\]
$\therefore $ The lowest term of the expression $\dfrac{{{x}^{2}}-4x-5}{{{x}^{2}}-25}\div \dfrac{{{x}^{2}}-3x-10}{{{x}^{2}}+7x+10}$ is \[\Rightarrow \dfrac{\left( x+1 \right)}{\left( x-5 \right)}\] .
Therefore, the correct option is (a).

Note: The biggest mistake that is common is dividing the final expression and correctly interpreting the divide sign. No confusion should occur while simplifying the division expression. For simplicity, a student may even write the divide expression as a division over division or inverse of multiplication. Taking this approach reduces the chances of error.